Question

O valor da integral ∫_1^e▒〖ln⁡x/x dx〗 é:

Answer 1

$\displaystyle \int\limits^e_1 { \frac{ln(x)}{x} } \, dx \\ \\ \\\text{ Fazendo a substituicao de variavel} \\ \\ u=ln(x) \\ du= \frac{1}{x}dx \\ \\ \\ \displaystyle \int\limits^e_1 u \, du \\ \\ \\ = \frac{u^1^+^1}{1+1} |^e__1$

$\displaystyle = {\frac{u^2}{2}|^e__ 1 }$

$\text{Substituindo o "u" } \\ \\ \\ = \frac{ln^2(x)}{2} |^e__1$

$=\frac{ln^2(e)}{2} - \frac{ln^2(1)}{2} \\ \\ \\ ln^2(e)=1 \\ ln^2(1)=0 \\ \\ \\ = \frac{1}{2}-0 \\ \\ =\boxed{ \frac{1}{2} }$

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Answer 2

0,5

Resolução do amigo esta correta.